It is shown that the matricesB k generated by any method from the restricted β-class of Broyden converge, if the method is applied to the unconstrained minimization of a functionf?C 2(R n ) with Lipschitz continuous? 2 f(x) and if the method is such that it generates vectorsx k converging sufficiently fast $$(\sum\limits_k {||x_k - x^* ||< \infty } )$$ to a local minimumx * off with positive definite? 2 f(x *). This result not only holds for constant step sizes? k ?1 in each iterationx k ?x k+1=x k ?? k B k ?1 ?f(x k ) of these methods but also for step sizes determined by asymptotically exact line searches. The paper generalizes corresponding results of Ge Ren-Pu and Powell [6] for the DFP and BFGS methods used in conjunction with step sizes? k ?1.